Abstract: Abridged The relatively recent insight that energy input from supermassiveblack holes BHs can have a substantial effect on the star formation ratesSFRs of galaxies motivates us to examine its effects on the scale of galaxygroups. At present, groups contain most of the galaxies and a significantfraction of the overall baryon content of the universe. To explore the effectsof BH feedback on groups, we analyse two high resolution cosmological hydrosimulations from the OverWhelmingly Large Simulations project. While bothinclude galactic winds driven by supernovae, only one includes feedback fromBHs. We compare the properties of the simulated groups to a wide range ofobservational data, including hot gas radial profiles and gas mass fractionsfgas, luminosity-mass-temperature L-M-T scaling relations, K-bandluminosity of the group and its central brightest galaxy CBG, SFRs and agesof the CBG, and gas-stellar metallicities. Both runs yield entropy profilessimilar to the data, while the run without AGN feedback yields highly peakedtemperature profiles, in discord with the observations. Energy input from BHssignificantly reduces fgas for groups with masses less than ~10^14 Msun,yielding fgas-T and L-T relations that are in agreement with the data. The runwithout AGN feedback suffers from the well known overcooling problem; theresulting K-band luminosities are much larger than observed. By contrast, therun that includes BH feedback yields K-band luminosities and CBG SFRs and agesin agreement with current estimates. Both runs yield very similar gas-phasemetallicities that match X-ray data, but they predict very different stellarmetallicities. Based on the above, galaxy groups provide a compelling case thatBH feedback is a crucial ingredient in the formation of massive galaxies.